Radiation from Transmission Lines PART I: Free Space

Home , Directivity, Electrical length, Radiation pattern, Radiation resistance

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radiate and rather carries power in the z direction only, but a semi-infinite TL does radiate, and we show in this section the connection between the finite and semi-infinite case and how the transition between them occurs. Here, it is important to mention that in some senses a finite matched TL is very similar to an infinite TL: in both cases there is no reflected wave, but they are very different in what concerns radiation: the first radiates and the second does not. In Section V we discuss the radiation resistance and generalize the formula derived in [4]. The work is ended with some concluding remarks. Note: through this work, the phasor amplitudes are RMS Fig. 2. The transmission line is modeled as a twin lead in free space, with values, hence there is no 1/2 in the expressions for power. distance d between the conductors. The currents in the transmission line flow Also, it is worthwhile to mention that the results of this work in the z direction at x = ±d/2 and they contribute to the magnetic vector depend on physical sizes relative to the wavelength, and hence potential Az. The termination currents (source or load) flow in the x direction and contribute the magnetic vector potential Ax. The arrows on the conductors are valid for all frequencies. show the conventional directions of those currents. The wires appear in the figure with finite thickness, but are considered of 0 radius. The transmission line goes in the z direction from z1 to z2. II. POWER RADIATED FROM A FINITE TL A. Matched TL presented as “traveling wave antenna” in many antenna and We calculate in this subsection the power radiated form a electromagnetic books like [9]–[13]. The last ones consider TL of length 2L, carrying a forward wave, represented by the only the current in the “upper” conductor in Figure 2, while current we consider all 4 currents appearing in the figure. This is I(z)= I+e−jkz (1) not an attempt to criticize those works, but only to mention We set z = L and z = L in the expression for their results do not represent radiation from transmission lines, 1 2 the z directed magnetic− vector potential in Eq. (A.13) from hence are not comparable with the results of this work or [3]– Appendix A, and obtain in spherical coordinates: [7].

It should be also mentioned that power loss from TL is Az = µ0G(r)F(z)(θ, ϕ) (2) also affected by nearby objects interfering with the fields, line bends, irregularities, etc. This is certainly true, but those where G is the 3D Green’s function defined in (Eq. (A.2)), affect not only the radiation, but also the basic, “ideal” TL and model in what concerns the characteristic impedance, the F (θ, ϕ)= jI+2Lkd sin θ cos ϕ sinc [kL(1 cos θ)] (3) propagation wave number, etc. Those non-ideal phenomena (z) − are not considered in the current work, nor in [3]–[7], and is the directivity function, and the subscript (z) denotes the also not in [9]–[13]. contribution from the z directed currents. The sinc function The methodology we use for calculating radiation losses is defined sinc(x) sin x/x. To obtain the far fields (those is first order perturbation: we use the lossless (0’th order decaying like 1/r),≡ the ∇ operator is approximated by jkr solution) for the electric current to derive the losses, and and one obtains: − we therefore use in Appendix A the e−jkz dependence. This b H 1 ∇ z methodology is used to derive the ohmic and dielectric losses (z) = (Az )= jkG(r)F(z)(θ, ϕ) sin θϕ (4) µ0 × [10]–[12], [14], and the same approach is used in different b b radiation schemes from free electrons: one uses the 0’th order and E(z) = η0H(z) r, where η0 = µ0/ǫ0 = 120π Ω is the × current (which is unaffected by the radiation) to calculate the free space impedance. For the contributionp of the x directed radiation [17]–[19]. To be mentioned that the same approach end currents we sumb the results for the x directed magnetic has been used in [3]–[7] (although [4] discussed about higher vector potential from Eq. (A.16) in Appendix A (setting z1 = order terms, without applying them). L and z2 = L), obtaining − The main text is organized as follows. In Section II we use A = µ G(r)F (θ, ϕ) (5) the results of Appendix A to calculate the power radiated from x 0 (x) a finite TL carrying a forward wave current, and generalize this where result for any combination of waves. As mentioned earlier, the + F(x)(θ, ϕ)=2jI d sin [kL(1 cos θ)] (6) results of Appendix A are applicable also for semi-infinite TL, − but we prefer to start with the finite TL, because in the follow- is the directivity function, and the subscript (x) denotes the ing Section III we validate the analytic results of Section II by contribution from the x directed currents. The fields from the comparing them with ANSYS commercial software simulation x directed end currents are calculated, obtaining results, and with published results obtained by other authors. H 1 ∇ x Section IV we base on Appendix A to analyze an infinite (x) = (Ax )= jkG(r)F(x)(cos θ cos ϕϕ+sin ϕθ) µ0 × − and semi-infinite TL. As expected, an infinite TL does not (7) b b b 3

+ and E(x) = η0H(x) r. Now summing the ﬁelds contributed The function D is 0 for θ = 0, and its number of lobes by the z directed currents× with those contributed by the x increases as the TL length increases. A one dimensional plot b + directed currents, we obtain H = H(z) + H(x): of D as function of θ for different TL lengths is shown in Figure 4. H = jkG(r)[ϕ(F(z) sin θ F(x) cos θ cos ϕ) θF(x) sin ϕ]. − − (8) b b 2 Using Eqs (3) and (6), the explicit expression for the far λ/4 λ/2 magnetic ﬁeld is 3λ/4 λ H+ = G(r)2kdI+ sin 2kL sin2(θ/2) [ϕ cos ϕ θ sin ϕ]. − − 1.5 (9) b We now use the superscript + on all the quantitiesb calculated ) θ in this subsection, to denote that they refer to a forward wave. (

+ 1 The far electric ﬁeld is D

+ + E = η0H r (10) × 0.5 We remark that the polarization of the fieldsb is not well defined at θ =0 (as shown in Figure 3), but this is not a problem in this case, because the fields are 0 at this point. 0 0 0.2 0.4 0.6 0.8 1 θ/π

Fig. 4. D+ as function of θ for TL 2L = λ/4, λ/2, 3λ/4 and λ.

The radiated power relative to the forward wave propagating power (P + = I+ 2Z ) is given by | | 0 P + 60Ω rad 2 (15) + = (kd) [1 sinc(4kL)] , P Z0 − As results from Eq. (15), the calculation of the relative radiated power requires the knowledge of two parameters: the separation distance in the twin lead representation d (relative to the wavelength), and the characteristic impedance of the actual cross section Z0, and in Appendix B we show examples of how to calculate this two parameters for different cross sections. In the following subsection we generalize the radiation losses for a TL with any termination. b Fig. 3. The polarization of the H+ field ϕb cos ϕ − θ sin ϕ , according to Eqs. (9), around θ = 0 (“north pole”). The z axis comes toward us from the center of the plot (at θ = 0), ϕ is 0 at the right side and increases counterclockwise. The polarization is not defined at θ = 0, but the fields are B. Generalization for non matched line 0 at this location. We generalize here the result (13) obtained for the losses The far Poynting vector S+ = E+ H+∗ comes out of a finite TL carrying a forward wave to any combination of × waves, as follows: rη k2d2 I+ 2 S+ = 0 | | sin2 2kL sin2(θ/2) (11) 4π2r2 I(z)= I+e−jkz + I−ejkz (16) b so that the total radiated power is calculated via where I+ is the forward wave phasor current, as used in 2π π − sin θdθdϕr2r S. (12) the previous subsection and I is the backward wave phasor Z0 Z0 · current, still defined to the right in the “upper” line in Figure 2. and comes out b The solution for a backward moving wave (only) on the finite TL, with a current phasor amplitude I− can be found P + = 60Ω I+ 2(kd)2 [1 sinc(4kL)] . (13) by first solving for a reversed z axis in Figure 2, i.e. a z axis rad | | − going to the left, replacing in the solution I+ I−. But The radiation pattern function is calculated from the radial this defines exactly the same configuration for the→ − backward pointing vector (Eq. 11) and the total power in Eq. (13): D+ = wave, as the original z axis defined for a forward wave, hence 4πr2S+/P + , which comes out r rad resulting in the same solution in Eqs. (9) and (10). Now to sin2[2kL sin2(θ/2)] express the solution for the backward wave in the original D+(θ)=2 . (14) [1 sinc(4kL)] coordinates, defined by the right directed z axis, one has to − 4 replace: θ π θ, ϕ ϕ, and therefore also θ θ and where Γ ZL−Z0 . Using (22), the radiation pattern is ZL+Z0 ϕ ϕ.→ Hence,− for a→− backward wave, the far H→− field is ≡ →− b b A2 + Γ 2B2 2AB cos(2ϕ) Γe−2jkL − − 2 D(θ, ϕ)=2 | | − ℜ{ }, Hb = bG(r)2kdI sin 2kL cos (θ/2) [ϕ cos ϕ + θ sin ϕ], 2 − (1 + Γ ) [1 sinc(4kL)] (17) | | − (23) b b so that the polarization at θ = π is not defined, and it looks where A and B are abbreviations for: like in Figure 3, this time the center of the plot is θ = π, and 2 2 increases clockwise. Again, this is no problem because for A sin 2kL sin (θ/2) B sin 2kL cos (θ/2) (24) ϕ ≡ ≡ this case the fields vanish at θ = π. For a matched TL, Γ=0, and Eq. (23) reduces to Eq. (14). For a backward wave only, the radiation pattern is: We remark that although the interference between the forward sin2[2kL cos2(θ/2)] and backward waves does not contribute to the radiated power D−(θ)=2 , (18) [1 sinc(4kL)] (see Eq. 21), it distorts the radiation pattern and introduces a − ϕ dependence. which looks like in Figure 4, only reflected around θ = π/2. As mentioned in the introduction, the “traveling wave We sum the fields of the forward and backward waves given antenna” presented in [9]–[13] do not represent transmission in Eqs. (9) and (17), obtaining lines, and therefore the radiation patterns in (14) or (23) are not comparable with those presented in the above references. H = kG(r)2d They will be however compared with [6], [15], [16]. [ ϕ cos ϕ(I+ sin 2kL sin2(θ/2) + I− sin 2kL cos2(θ/2) )+ In the next section we validate the analytic results obtained − θ sin ϕ(I+ sin 2kL sin2(θ/2) I− sin 2kL cos2(θ/2) )], in this section, using ANSYS commercial software simulation b − (19) and additional published results on radiation losses from TL. b so the far Poynting vector is III. VALIDATION OF THE ANALYTIC RESULTS 4(kd)2 S =rη I+ 2 sin2 2kL sin2(θ/2) + I− 2 A. Comparison with ANSYS simulation results 0 16π2r2 | | | | 2 2 We compare in this subsection the relative radiated power bsin 2kL cos (θ/2) + 2cos(2ϕ) from a two-conductor TL (Eq. 15) carrying a forward wave, sin 2kL sin2(θ/2) sin 2kL cos2(θ/2) I+I−∗ ℜ{ } with the relative radiation losses results obtained from ANSYS (20) commercial software simulation. For the comparison we use The interference between the waves does not contribute to the the cross section of two parallel cylinders, for which the 2π analytic solution is known from image theory [10]–[12], [14], radiated power (because 0 dϕ cos(2ϕ)=0). The radiated power comes out R but also confirmed by Appendix B. The cross section is shown in Figure 6. The diameters of the cylinders are 2a =0.0203λ, P =60 Ω(kd)2 I+ 2 + I− 2 [1 sinc(4kL)] and the distance between their centers is s =0.02872λ, where rad | | | | − ≡ P + + P − , (21) the wavelength λ =6.25 cm, corresponding to the frequency rad rad of 4.8 GHz. The distance between the image currents (shown + − where Prad is given in Eq. (13) and Prad is similar, only replace I+ by I−. The radiation pattern for the general case of forward and 2 backward waves is D = 4πr Sr/Prad, where Sr is given in Eq. (20) and Prad in (21). To express D independently of the currents, it is convenient to consider a general TL circuit in Figure 5, for which the relation between I+ and I− is

Fig. 5. TL fed by a generator VG with an internal impedance ZG, loaded by ZL. The value of ZL affects indirectly the radiated power by setting the Fig. 6. Cross section of two parallel cylinders: the distance between the + − relation between the forward and backward currents I and I . centers is s = 0.02872, and the diameters are 2a = 0.0203 wavelengths. The the red points show the current images which define the twin lead representation, and the distance between them d = 0.0203 wavelengths is I−ejkL +ΓI+e−jkL =0, (22) calculated in Eq. (25). 5 as red points in Figure 6) is the separation distance d in the The real part of Θ represents the phase accumulated by a twin lead model, given by forward wave along the TL, and the imaginary part of Θ (which is always negative) represents the relative decay of the d = s2 (2a)2 =0.0203λ, (25) forward wave (voltage or current) due to losses (in our case p − so that (kd)2 =0.016 is small enough, and the characteristic there are only radiation losses) along the TL, so that I+(L) = impedance is I+( L) exp(Im Θ ). Therefore, the power carried| by| the | − | {+ } + η0 d + s forward wave P (L) = P ( L) exp(2Im Θ ), but for Z = ln = 105.6Ω (26) |+ | +| − | { } 0 π 2a small losses P (L) P ( L) (1+2Im Θ ), so that the difference between| the| ≃ input | and− output| values{ } of P + (which Both analytic results for d and Z compare well with those 0 represent the radiated power P + in Eq. (15)), relative to the calculated in Appendix B for this cross section. rad (average) power P + carried by the wave is obtained by We simulated the configuration in Figure 6 using ANSYS- HFSS commercial software, in the frequency domain, FEM P + rad = 2Im Θ , (30) technique. The box surface enclosing the device constitutes a P + − { } radiation boundary, implying absorbing boundary conditions where Im is the imaginary part and Im Θ < 0 always. (ABC), used to simulate an open configuration that allows In Figure 8 and Table I we compare{ the} analytic result in waves to radiate infinitely far into space. ANSYS HFSS Eq. (15) with the result obtained from simulation Eq. (30), for ABC, absorbs the wave at the radiation boundary, essentially a fixed frequency of 4.8 GHz and different TL lengths. We see ballooning the boundary infinitely far away from the structure. The enclosing box surface has to be located at least a quarter 0.012 wavelength from the radiating source. For the frequency of Theory 4.8 GHz we used, the wavelength is 6.25 cm , and we chose the Simulation box sides 7.5 cm in the x and y directions, and the TL length 0.01 plus 2.5 cm on each side in the z direction. For the interface to the device we used lumped ports, which define perfect H 0.008

boundaries everywhere on the port plane, so that the E ﬁeld + on the port plane (outside the conductors) is perpendicular to /P 0.006 the conductors. rad P The simulation setup is shown schematically in Figure 7. 0.004 The TL is ended at both sides by lumped ports of characteristic impedance Zport = 50Ω, but fed only from port 1 by forward + 0.002 wave voltage Vport =1 V , so the equivalent Th´evenin feeding circuit is a generator of 2V + in series with a resistance Z . port port 0 0 0.5 1 1.5 2 TL length [wavelengths]

+ Fig. 8. Relative radiation losses for a matched TL Prad/P : comparison between the analytic result in Eq. (15) and the simulation result in Eq. (30) for different TL lengths in units of wavelengths.

TABLE I THE NUMERICAL DATA FROM FIGURE 8 AND THE RELATIVE ERROR. Fig. 7. Simulation setup for obtaining 2 × 2 S matrices for different TL lengths. TL length simulation (Eq. 30) theoretical (Eq. 15) % error 0.08 0.001963 0.001479 32.76 We obtained from the simulation S matrices deﬁned for a 0.16 0.005796 0.005079 14.12 characteristic impedance of at both ports (which is an 0.24 0.009502 0.008851 7.35 Zport 0.4 0.011115 0.010982 1.21 arbitrary choice), for different lengths of the transmission line. 0.6 0.007987 0.008069 -1.02 By symmetry, the S matrix has the form 0.8 0.009878 0.009774 1.06 0.96 0.009575 0.009603 -0.28 Γ τ S = , (27) 1.12 0.008198 0.008579 -4.44 τ Γ 1.2 0.008646 0.008874 -2.57 1.28 0.009366 0.009445 -0.84 from which one may calculate the ABCD matrix of the TL 1.44 0.009589 0.009583 0.05 [14], [20]–[22]. We need only the A element from the matrix: 1.6 0.008665 0.008797 -1.50 1 1.76 0.009216 0.009286 -0.75 A = τ + (1 Γ2)/τ (28) 1.92 0.009746 0.009557 1.98 2 − 2.08 0.009046 0.008936 1.23 from which we compute the delay angle (or electrical length) of the TL that the simulation conﬁrms well the theoretical result, with an Θ = arccos(A) (29) average absolute relative error of 4.75%. The biggest relative 6

errors are at the short TL, where the relative radiation losses where Prad is given in Eq. (21). Using Eqs. (22), (34) and (35) are low, and hence more difficult to reproduce accurately with we obtain the simulation. For example, if we exclude the shortest TL 2 2 1+ Γ length of 0.08λ from the comparison, the average absolute rrad = 60Ω(kd) [1 sinc(4kL)] | | , (36) − 1 Γe−4jkL 2 relative error is 2.75%, or if we exclude the 2 shortest points | − | of 0.08 and 0.16λ from the comparison, it drops to 1.87%. which is identical to Eq. (5) in [4], after setting the ohmic 1 1+x attenuation α to 0, and use the identity arctan(x)= 2 ln 1−x and the definition of the cosh function. B. Comparison with [3] One remarks that rrad in Eq. (36) goes to infinity if Γ =1 | | In 1923 Manneback [3] published a paper “Radiation from and 4kL 6 Γ=2πn (n integer). This lacuna will be fixed − transmission lines” which calculated the power radiated by two in Section V. thin wires of length l (equivalent to our 2L), and separation d, In the private case of a matched TL, using the radiated in resonance, having open terminations. The author considered power in Eq. (13) divided by I+ 2, or alternatively setting | | the current Γ=0 in Eq. (36) results in:

Im cos(knz) sin(ωnt) (31) 2 rrad = 60Ω(kd) [1 sinc(4kL)] , (37) − where kn = πn/l and ωn = ckn for odd n (as defined which is identical to the case shown in Eq. (6) in [4], after in Eq. 3 of [3]). Note that the time dependence has been setting the attenuation α to 0. explicitly written in [3] and the calculations have been done in time domain, but using a fixed frequency, hence they are completely equivalent to our phasor calculations. The result D. Comparison with [5] for the radiated power is given in Eq. 11 of [3], rewritten here Another comparison is with Bingeman’s work from 2001 for convenience [5], in which the method of moments (MoM) has been used to calculate the radiation from two thin wires of diameter 2a = P = 15Ω(kd)2I2 (32) rad m 5 mm, length 2L = 10 m and separated at a distance of d = We compare this with our result (21). kl = nπ is in our 1 m. The characteristic impedance is given by Eq. (26) (but given d a one may use s = d, see Eq. (25)) and results in notation kL = nπ/2 hence the sinc function in Eq. (21) results ≫ 0. This resonant case implies I+ = I−, so the general current Z0 = 720Ω, as calculated at the beginning of [5]. In absence in Eq. (16) reduces to of other losses, the author derived the radiated power as the difference between the power carried by the TL and the power 2I+ cos(kz) (33) reaching the load. The first calculation is the power radiated by a matched and Eq. (21) results in 120 Ω(kd)2 I+ 2. | | TL, fed by a power of 1000 W, for frequencies f = 2, The value Im in Eq. (31) is the amplitude of the current, i.e. 5, 7, 10, 15 and 20 MHz. For the power of 1000 W, the the RMS value times √2. We use RMS values (as mentioned RMS value of the forward current to set in Eq. (13) is in the introduction) hence the equivalence between Eq. (31) + I = 1000/Z0 =1.1785 A, and we use k = 2πf/c for and Eq. (33) is by setting I = 2√2 I+ , and using this | | m | | the abovep frequencies. The numerical results for this case are equivalence, Eq. (21) reduces exactly to Eq. (32). given in Table 1 of [5], and we compare those results to ours, To be mentioned that our results are general, covering all in Figure 9 and Table II. We see a good match for the high cases of terminations, or any combination of waves, and we showed in this subsection how our general result reduces TABLE II correctly to the result for a private case of resonance. THE NUMERICAL DATA FROM FIGURE 9 AND THE RELATIVE ERROR.

Frequency [MHz] [5] theoretical (Eq. 13) % error C. Comparison with [4] 2 0 0.0165 -100 5 1 0.5359 87 In 1951 J. E. Storer and R. King, published the paper 7 2 1.664 20 “Radiation Resistance of a Two-Wire Line”, in which they 10 4 4.411 -9.32 15 8 8.225 -2.73 calculated the radiation resistance of a twin lead TL loaded by 20 13 13.11 -0.84 an arbitrary load, i.e. carrying an arbitrary combination of forward and backward waves, shown schematically in Figure 5. frequencies (big electric delay), and it deteriorates at small Z −Z0 Defining the complex reflection coefficient Γ L , the electric delays. But the result 0 for the frequency of 2 MHz ≡ ZL+Z0 relation between I+ and I− is given in Eq. (22). The current is clearly incorrect, so we may understand that the accuracy at the generator side is: of the results in [5] is low at small electric delays, for which the relative radiated power is small. I( L)= I+ejkL + I−e−jkL, (34) − Another calculation in [5] is for a non matched TL, with and the radiation resistance is defined by end loads RL=10, 50, 500 Ω, 1, 5, 10, and 50 kΩ, all cases at frequency 10 MHz, carrying a net power of 1000 W. We 2 rrad Prad/ I( L) , (35) compare those results with the results of Eq. (21). First k = ≡ | − | 7

TABLE III 14 Computed analytically THE NUMERICAL DATA FROM FIGURE 10 AND THE RELATIVE ERROR. Numerical results from [5] 12 Load RL [Ω] [5] theoretical (Eq. 21) % error 10 140 158.83 -11.85 10 50 32 31.91 0.28 500 5 4.70 6.38 1k 5 4.65 7.53 8 5k 15 15.63 -4.03

[W] 10k 29 30.79 -5.81

rad 6 50k 128 153.19 -16.44 P 4 approaches the order of magnitude of the net power (1000 W). 2 This may be due to the limitations of the current theory to small losses that almost do not affect the basic electromagnetic 0 2 4 6 8 10 12 14 16 18 20 solution (see Introduction). frequency [MHz] E. Comparison with [6] Fig. 9. Comparison between the radiated power for a matched line from Table 1 of [5] with our results for a matched line in Eq. (13) for different In 2006 Nakamura et. al. published the paper “Radiation frequencies. Characteristics of a Transmission Line with a Side Plate” [6] which intends to reduce radiation losses from a twin lead TL using a side plate. The side plate is a perfect conductor put 2πf/c =0.2094 [1/m] is ﬁxed, and we calculate for each load aside the transmission line, to create opposite image currents, resistance RL and hence reduce the radiation. R Z Γ = L − 0 , (38) The authors ﬁrst derived the radiation from a TL without | | R + Z L 0 the side plate, obtaining an integral (Eq. 20 in [6]) which from which the forward power values for each case are given they computed numerically. The numerical integration result by is shown in Figure 6 of [6], where the solid line represents P 1000 P + = = . (39) the free space case. 1 Γ 2 1 Γ 2 We compare our analytic result in Eq. (13), with the − | | − | | numerical result shown in Figure 6 of [6]. First, in [6] The forward current values for each case are given by I+ = I0 + | | is a forward current, and from Eq. (19) in [6], it is evident P /Z0 and the backward current values for each case are − + that they used RMS values. They used I0 = 1A, hence we pgiven by I = Γ I . Setting the values in Eq. (21), we + compare the| results| | || of [5]| for the unmatched line at 10MHz set I = 1A in Eq. (13). 2h is the distance between the conductors| | in [6], equivalent to d in this work, and they (Table 2 in [5]), with the results of Eq. (21) in Figure 10 and 2 2 Table III. The match between the results is good, except for used hλ = 0.1, therefore (kd) = (4πh/λ) = 1.5791 in Eq. (13), so that the total radiated power for the case displayed in Figure 6 of [6] is

160 Computed analytically P =60 1 1.5791 [1 sinc(4kL)] = Numerical results from [5] rad × × − 140 2L 94.746[W ] 1 sinc 4π , (40) λ 120 −

100 and we wrote the argument of the sinc function in terms of 2L/λ, i.e. the TL length in wavelengths. This result is [W] 80 displayed in Figure 11, in which we show the radiated power rad P 60 as function of the TL length in wavelengths. The authors did not supply the numerical data to reproduce Figure 6 of [6], 40 and we did not want to copy the ﬁgure into this work for 20 comparison, but we checked very carefully that indeed our calculation shown in Figure 11 completely overlaps the solid 0 101 102 103 104 line in Figure 6 of [6]. Ω We compare as well the radiation patterns obtained in [6] RL [ ] with ours (Eq. 14) in Figure 12. We remark that D+(θ) is not symmetric around θ = π/2 in general (see Figure 4), but Fig. 10. Comparison between the radiated power for a non matched line from for the cases kL = nπ/2 (integer n), i.e. the TL length is a Table 2 of [5] with our results for a non matched line in Eq. (21) for different load resistances. multiple integer of half wavelength, displayed in Figure 12, D+(θ) is symmetric around θ = π/2, because sin2(nπ/2+ the ﬁrst and last cases, in which the radiated power is big and x) = sin2(nπ/2 x) for any x. The radiation patterns in − 8

120

100

[W] 60 rad P 40 Fig. 13. Transmission line of length λ/4, open at one termination, and shorted at the other, represents a “U” shaped antenna. 20 kL = π/4 the radiation pattern in Eq. (23) is: 0 2 2 2 2 0 1 2 3 4 5 D = sin (π/2) sin (θ/2) + sin (π/2)cos (θ/2) =1. TL length [wavelengths] (41) Using (π/2)cos2(θ/2) = (π/2) (π/2) sin2(θ/2) one eas- Fig. 11. Recalculation of the solid line in Figure 6 of [6], using Eq. (13). ily remarks that (41) is 1, describing− uniform radiation, as mentioned in [15], [16]. This does not contradict the “hairy- TL length = 0.5λ TL length = λ ball” theorem [23], because this theorem states that any real tangential field must be 0 at least at one point on a sphere, and by real one means: having the same phase everywhere. x x And indeed the separate fields associated with I+ and I− (each one “real” in the above sense) given in Eqs. (9) and z y z y (17) are 0 at points θ =0 and π respectively. From Eq. (22), the relation between the forward and backward wave is I− = + TL length = 1.5λ TL length = 2λ jI , and after summing the fields by setting this relation −into Eq. (19), the far magnetic field is H = G(r)2kdI+ x x 2 2 [ ϕ cos ϕe−j(π/2) cos (θ/2) + θ sin ϕej(π/2) cos (θ/2)], (42) − z y z y whichb has a constant amplitudeb everywhere, but a changing phase, which cannot be factored out to get a “real” field. This complex field manifests a linear polarization at θ =0 and π, Fig. 12. Radiation pattern D+ calculated from Eq. (14) for the cases of TL lengths nλ/2, for n=1 to 4. They are identical to the parallel cases shown in circular polarization at θ = π/2 and ϕ multiple of π/4, and Figure 5, panel(a) of [6]. Note that the definitions of the x and z axes are elliptic elsewhere, compare with [15], [16]. swapped in [6] compared to our definitions, we therefore showed them in an orientation which makes the comparison easy (i.e. our z axis is oriented in IV. INFINITE AND SEMI-INFINITE TL ANALYSIS the plots in the same direction as their x axis). We remark that the pattern for the case 0.5λ is quite similar to this of a dipole antenna, because a short As we know, there are no infinite or semi-infinite TL in TL behaves similar to a small magnetic loop, i.e. a magnetic dipole. reality, but the literature considers those kind of TL as limiting cases, and as we shall see, the analysis of the infinite and semi- infinite TL supplies an additional insight and validation of the Figure 12 are identical to the parallel cases shown in Figure 5, results obtained in the Section II, as shown in the following panel(a) of [6]. subsections. It is worthwhile to remark that the radiation pattern (Eq. 14) Certainly those cases can be considered only in absence of does not depend on the distance between the conductors d (or other losses, like ohmic or dielectric, for which infinite TL 2h in [6]), hence the annotation of h/λ = 0.1 in Figure 5 have infinite losses. As one remarks, the radiation losses of of [6] is redundant, and probably has been added to the finite TL reach an asymptotic value for long TL, so that one caption because the authors computed the radiation patterns may expect that infinite or semi-infinite TL do not radiate an numerically for h/λ = 0.1, without deriving an analytic infinite power. expression. A. Infinite TL For an infinite TL, carrying a forward wave, we set z = L F. Comparison with [15], [16] 1 and z = L in the result (A.13) and considering L (i.e.− 2 → ∞ References [15], [16] analyze the radiation from a “U” for finite z and ρ, L z ,ρ) we obtain ≫ | | + −jkz shaped antenna (see Figure 13) and showed that its radiation µ0I 2e pattern is uniform. Using Γ = 1 (shorted termination) and Az = d cos ϕ . (43) − 4π ρ 9

Certainly, we do not have in this case x directed currents, so and from Eq. (A.16) for z2 =0: we obtain from Eq. (43): −jkr + e 1 e−jkz I+d Ax RT = µ0I d , (51) H = ∇ A = [ ρ sin ϕ + ϕ cos ϕ] (44) − 4πr µ 2π ρ2 0 × − while for the second case we set z1 =0, z2 = L in Eq. (A.13) and b b and taking L we obtain → ∞ 1 e−jkz I+d + −jkr E ∇ H ρ ϕ (45) µ0I ρ e = = η0 2 [ cos ϕ + sin ϕ], Az LT = d cos ϕ (52) jωǫ0 × 2π ρ 4π r z r − which are the static H and E fields multipliedb byb the forward and from Eq. (A.16) for z1 =0: wave propagation factor e−jkz. −jkr We remark that in Appendix A we considered the far field + e Ax LT = µ0I d , (53) (kρ 1), so the fields in Eqs. (44) and (45) are correct far 4πr ≫ from the TL, and their diverging at ρ =0 is an artifact of this where the subscripts RT and LT mean “right terminated” far field approximation. But even in the far field, writing them and “left terminated” TL, respectively. Looking at the RT in spherical coordinates so that ρ = r sin θ, the fields decay like 1/r2 and there is no “radiating” term decaying like 1/r. This is also evident from the Poynting vector: 1 I+ 2d2 S = E H∗ = η | | z, (46) × 0 4π2 ρ4 which is only in the z direction, representing theb power carried by the TL. Given the fact that the fields in Eqs. (44) and (45) decay much faster than radiating fields, hence are negligible relative to them far from the TL, it is convenient to define a typical distance ρ0 from the TL, so that

Static near ﬁeld is dominant if ρ<ρ0

Radiation field is dominant if ρ>ρ0. (47) There are no radiation fields in this subsection, but the relation (47) will be referred to in the next subsection, analyzing semi- Fig. 14. Semi-infinite TL from z = −∞ to the center of coordinates at z = 0. The blue circle represents the wave front of the outgoing spherical infinite TL. wave radiation in the second part of Eq. (50) and the red wave fronts represent Another way of understanding ρ0 is by integrating the the near plane wave field in the first part of Eq. (50). The near plane wave and Poynting vector to obtain the forward power P + spherical wave cancel each other in the paraxial region z > 0 and ρ < ρ0, see Eq. (54). The spherical wave is shown dashed blue in the canceling region, ∞ which occurs within a cone ∆θ = ρ0/r (dashed black line). The cone gets + P = dxdy S z, (48) narrower as the distance from the center of coordinates r increases. ZZ−∞ · and here one has to use the exact fields inb the expression for S configuration in Eq. (50), we see that it includes also the near (not the far fields in Eqs. (44) and (45)). To obtain P + with a field expression of the infinite TL from Eq. (43) for all z “reasonable” required accuracy, one does not need to integrate in spite of the fact that the TL is in the region z < 0 and terminates at z = 0. This is explained by the fact that for to infinity, but rather 2 ρ 2π ρ0 z > 0 and small ρ (typically ρ<ρ0, see (47)), r z + 2z , 2 ≈ P + dϕ dρ ρ S z, (49) ρ so that r z 2z , resulting in ≃ Z0 Z0 · − ≈ b ρ e−jkr 2e−jkz so that ρ0 is the radial distance from the TL (in cylindrical , (54) coordinates) within which the near fields are significant. r z r z>0 ≃ ρ ρ<ρ0 − which means that the spherical wave in the second part of B. Semi-infinite TL Eq. (50) describes radiation everywhere except in a cone We analyze here a semi-infinite TL carrying a forward wave. around θ = 0 where it is canceled by the near plane wave The TL can be either from z = to 0 (Figure 14) or from in the first part of Eq. (50), see Figure 14. z =0 to (Figure 15). We note−∞ that in both cases we have ∞ For the LT configuration the spherical wave in Eq. (52) to consider also the contribution of the x directed current at represents radiation except inside a cone around θ =0, where the termination at z =0. it equals the near plane wave in the first part of Eq. (50) For the first case we set z = L, z = 0 in Eq. (A.13) 1 − 2 (according to Eq. (54)), which does not have radiating fields, and taking L we obtain see Figure 15. → ∞ + µ I+ 2e−jkz ρ e−jkr So to calculate the radiated power P , one may either A = 0 d cos ϕ (50) rad z RT 4π ρ − r z r use the second part of Eq. (50) together with Eq. (51), or − 10

resulting in Poynting vector E+ H+∗ : × (kd)2r S+ = 30Ω I+ 2, (63) 4πr2 | | b So that the total radiated power for a forward wave is

2π π + 2r S + 2 2 Prad = sin θdθdϕr = 30Ω I (kd) (64) Z0 Z0 · | | It is clear from Eqs. (61) andb (63) that this is an isotropic + 2 + + radiation. Calculating D =4πr Sr /Prad, one obtains D+ =1, (65)

so that we encounter again an isotropic radiation, but contrary to the case shown in Section III-F, here the polarization is linear. This is possible because the radiation field is not the Fig. 15. Semi-infinite TL from the center of coordinates at z = 0 to z = ∞. The blue circle represents the wave front of the outgoing spherical wave only field far from the origin, and the near plane wave is also radiation (Eq. (52)), but behaves in the paraxial region z > 0 and ρ < ρ0, present, see Figures 14 and 15. like the near plane wave in the first part of Eq. (50), according to Eq. (54). It is worthwhile at this point to understand the connection Therefore, in this paraxial region the spherical wave front (dashed blue) does not represent radiation, but rather the near plane wave. Like in Figure 14, this between the radiation of a finite TL and a semi-infinite TL. paraxial region is within the cone ∆θ = ρ0/r (dashed black line), which In reality, a semi-infinite TL is a very long TL, for which gets narrower as the distance from the center of coordinates r increases. we analyze the termination near to “our” side, while someone else analyzes the termination near to “his/her side”, as shown in Figure 16. Figure 17 shows schematically how the power Eq. (52) together with Eq. (53), excluding the paraxial region around θ = 0. This exclusion is meaningless from the point of view of the calculation, because as the distance from the center of coordinates r increases this region reduces to a singular point. The spherical potential vectors for the RT and LT configurations differ only by sign, so both yield the same + result for Prad. Using the RT configuration, we rewrite

Az = µ0F(z)(θ, ϕ)G(r), (55) where Fig. 16. Very long TL, carrying a forward wave, for which one considers each + sin θ termination as the termination of a semi-infinite TL. Each termination radiates F(z)(θ, ϕ)= I d cos ϕ , (56) − 1 cos θ the power 30 Ω|I+|2(kd)2, so that the whole TL radiates 60 Ω|I+|2(kd)2, − the asymptotic value in Eq. (13). is the directivity function. We calculate now the radiating electric and magnetic fields, i.e. the part of the fields which radiated by a TL carrying a forward wave, gradually changes decays like 1/r, by approximating ∇ jkr, obtaining ≃− as the TL length increases. H (z) = jkF(z)(θ, ϕ) sin θG(r)ϕb, (57) E H r and (z) = η0 (z) . Now we rewrite (51):b × Ax =b µ0F(x)(θ, ϕ)G(r), (58) where the directivity function F(x) is F (θ, ϕ)= I+d, (59) (x) − The fields are H = ( jkr) (A x)/µ (x) − × x 0 H = jk(cos θ cos bϕϕ + sinbϕθ)G(r)F , (60) (x) − (x) Fig. 17. A schematic diagram showing the connection between the radiation E H r H H of finite TL and semi-infinite TL carrying a forward wave. The red lines are and (x) = η0 (x) . Addingb up the fieldsb (z) + (x) we hand drawn and go around the element considered for the calculation of the obtain × radiation, so their shape is meaningless. Panel (a) shows a short TL radiating b the power 60 Ω(kd)2|I+|2 [1 − sinc(4kL)] according to Eq. (13). Panel (b) H+ = jkG(r)I+d[ϕ cos ϕ θ sin ϕ]. (61) shows a TL longer than several wavelengths, for which | sinc(4kL)| ≪ 1, − − which practically radiates 60 Ω(kd)2|I+|2, but still considered a single We named it H+, because it is the radiatingb field of a forward radiating element. Panel (c) shows a very long TL, like in Figure 16, which b is analyzed as two separate radiating elements (as shown in Figures 14 and wave. and 15), each one radiating 30 Ω(kd)2|I+|2, according to Eq. (64), in total the E+ = η H+ r (62) same as in panel (b). 0 × b 11

+ + + + + Till here we considered only a forward wave, and that is P (L) = P ( L) Prad, but considering Prad P , − − + + + ≪+ what one usually considers for a semi-inﬁnite TL from z =0 this may be written as P (L)= P ( L) 1 Prad/P , or + + − − + + to (LT case), but for the RT case terminated by a non more conveniently P (L)= P ( L) exp( Prad/P ). ∞ + − + −− − matched load, one can have both forward and backward waves. The small decay factor Prad/P (or Prad/P for the The generalization for this case is done like in Section II-B, backward wave) is minus twice the imaginary part of the TL and the total radiated power in presence of a forward and electrical length 2Im Θ according to Eq. (30), so we may backward wave is describe the dynamics{ of} P + or P − along the TL

2 + 2 − 2 + + 2Im{Θ} Prad = 30Ω(kd) ( I + I ), (66) P (L)= P ( L)e (67) | | | | − so that the interference between the waves does not contribute P −( L)= P −(L)e2Im{Θ} (68) to the radiated power, similarly to the case of a finite TL. − where Im Θ < 0 always. Given P ± are proportional to V. RADIATION RESISTANCE I± 2 respectively,{ } the forward and backward currents decay | | The radiation resistance has already been worked out in according to Im Θ , in addition to their accumulated phase, { } Section III-C, for comparison with [4]. It is defined by rrad = so we express 2 Prad/ I , Prad being the total power radiated by the TL, and | | I+(L)= I+( L)e−j2kLeIm{Θ}, (69) I the current at the generator side (in Figure 5 it is I( L)). − − The radiation resistance is given in Eq. (36) and is identical I−( L)= I−(L)e−j2kLeIm{Θ}, (70) to Eq. (5) in [4]. − At the load side I−(L) = ΓI+(L), so using Eqs. (69) and However, we remark that rrad in Eq. (36) goes to infinity if − Γ =1 and 4kL 6 Γ=2πn (n integer). For example if Z in (70), we express the total current near the generator I( L)= L + − − Figure| | 5 is (open− TL), Γ=1, so that I(L)=0. If the length I ( L)+ I ( L): ∞ − − of the TL 2L is a multiple integer of λ/2, also the current at Im I( L)= I+( L) 1 Γe−j4kLe2 {Θ} (71) the generator side I( L)=0. This is shown schematically in − − − − h i Figure 18, for n =1. This case represents resonance (infinite which for Im Θ 1 can be written: | { }| ≪ I( L)= I+( L) 1 Γe−j4kL(1+2Im Θ ) (72) − − − { } + + Using Eqs. (35), (21) and the relation 2Im Θ = Prad/P from Eq. (15) we obtain a more accurate,− explicit{ } expression for the radiation resistance 60Ω(kd)2 [1 sinc(4kL)] (1 + Γ 2) rrad = −4jkL − 2 | | 2 . 1 Γe 1 (60Ω/Z0)(kd) [1 sinc(4kL)] | − { − − (73)}| Fig. 18. Open ended transmission line, of length 2L = λ/2. The current is 0 at both TL ends. The radiation resistance in Eq. (36) (or Eq. (5) in [4]) fails If Γe−4jkL is far from 1, the last term in the denominator is in this case, resulting infinity. negligible, and one recovers the approximate Eq. (36). On the −4jkL VSWR) and current at generator side 0. This of course does other hand if Γe =1 (resonance), one obtains not mean that the generator does not have to compensate for 2Z2 r = 0 . (74) the radiated power, but rather a fail of the small radiation losses rad 60Ω(kd)2 [1 sinc(4kL)] approximation. In such case P + = I+ 2Z equals P − = − 0 which is big, because kd 1, but not infinite. For the special I− 2Z so that the net power carried| by| the TL P = P + 0 case described in Figure≪ 18, Γ=1 and 4kL = 2π, the sinc P| −|= 0, hence the radiated power P is infinitely bigger− rad function is 0, so that r reduces to 2Z2/[60 Ω (kd)2]. than the net power P transferred by the TL. rad 0 We derive here a more robust radiation resistance, valid for any TL configuration. We still assume P + P + (small VI. CONCLUSIONS rad ≪ relative losses), but the total radiated power Prad is allowed We derived in this work a general radiation losses model to be bigger than the net power P . Given the fact that the for two-conductors transmission lines (TL) in free space. We interference between the forward and backward waves does considered any combination of forward and backward waves not contribute to the radiated power (see (Eq. 21)), we may (i.e. any termination), and also any TL length, analyzing consider the separate loss of the forward or backward wave. infinite, semi-infinite and finite TL. + + − − The relation Prad/P in Eq. (15), equal also to Prad/P , One important finding is that the interference between is written as if P + would be a constant, but P + is only forward and backward waves does not contribute to the + + approximately constant for Prad P . Looking at the radiated power (Eq. (21)), which has been also validated by configuration in Figure 5, by conservation≪ of energy, the the comparisons with [3]–[5], [15], [16], in the sense that those + power radiated by a forward wave Prad must be the difference comparisons would have failed if Eq. (21) were incorrect. P +( L) P +(L), which is small relative to the individual This property allowed us to consider the separate losses values− of−P +( L) and P +(L). We may therefore express for the forward an backward wave for the calculation of the − 12 radiation resistance in Section V. This radiation resistance Changing variable z′′ = z′ z in Eq. (A.1), one obtains − reduces correctly far from resonance to this calculated in [4] 2 z −z ′′ −jkz ′′ −jkz (Eq. 36), but handles correctly the resonant case. Az = µ0e dz dc Kz(c)e G(R), (A.4) Another novelty of this work is the analysis of the semi- Zz1−z I infinite TL which clearly shows that the radiation from TL is redefining R = (x x′(c))2 + (y y′(c))2 + (z′′)2. For a mainly a termination effect. We found an isotropic radiation − − far observer, at distancep ρ x2 + y2 from the TL, so that ρ from the semi-infinite TL, which is possible due to the fact ≡ is much bigger than the transversep dimensions of the TL one that the radiation fields are not the only far fields, as shown in approximates R in cylindrical coordinates as Figures 14 and 15. The semi-infinite TL radiation results are ρ consistent with finite TL results, so that a very long TL can R r [x′(c)cos ϕ + y′(c) sin ϕ] , (A.5) be regarded as two semi-infinite TLs, as shown in Figure 17. ≃ − r Although previous works [3]–[7] considered exclusively the where r(z′′) (z′′)2 + ρ2. We keep for now everything in ≡ twin lead cross section, the formalism developed in this work cylindrical coordinates,p to be able to handle infinite or semi- is valid for any cross section. We showed this in Section III-A infinite lines. Using this in Eq. (A.4), one obtains by successfully comparing the analytic results with simulation ′′ ′′ z2−z −jk[z +r(z )] of ANSYS-HFSS commercial software for a parallel cylinders −jkz ′′ e Az =µ0e dz ′′ cross section (Figure 6), in which the radius was not small Zz1−z 4πr(z ) relative to the distance between the centers of the cylinders. ′ ′ jk(ρ/r)[x (c) cos ϕ+y (c) sin ϕ] dc Kz(c)e . (A.6) Appendix B explains how to calculate the parameters needed I to derive the radiation for any TL cross section. Some comments on the generalization of this research for We consider the higher modes to be in deep cutoff, so that kx′(c),ky′(c) 1, hence TL in dielectric insulator. The case of TL in dielectric insulator ≪ ′′ ′′ is solvable analytically, but much more involved than the free z2−z −jk[z +r(z )] −jkz ′′ e space case. The fact that the TL propagation wavenumber Az µ0e dz dc Kz(c) ≈ Z 4πr(z′′) I β is different form the free space wavenumber k by itself z1−z 1+ jk(ρ/r)[x′(c)cos ϕ + y′(c) sin ϕ] . (A.7) complicates the mathematics, but in addition it comes out that { } one needs to consider in this case also polarization currents, Separating the contour integral dc = dc + dc , where which further complicate the results. Radiation from TL in 1 2 c1,2 are the contours of the “upper”H andH “lower”H conductors dielectric insulator will be published separately, as Part II of respectively (see Figure 1), and using this study. + dc1Kz(c1)= dc2Kz(c2)= I (A.8) APPENDIX A I − I FAR VECTOR POTENTIAL OF SEPARATED so that the integral on each surface current distribution results TWO-CONDUCTORSTRANSMISSIONLINE in the total current, which we call I+, because it represents We show in this appendix that for the purpose of calculating a forward wave. Given that for a two-conductors TL there is the far fields from a two ideal conductor transmission line (TL) only one (differential) TEM mode, this current is equal, but in free space, having a well defined separation between the with opposite signs on the conductors. We may define the 2D conductors, as shown in Figure 1, one can use an equivalent vector ρ(c) (x′(c),y′(c)), from which one defines the vector ≡ twin lead, provided the separation is much smaller than the distance between the center of the surface current distributions wavelength. + d dc1Kz(c1)ρ(c1)+ dc2Kz(c2)ρ(c2) /I . (A.9) For simplicity we use a forward wave (propagating like ≡ I I e−jkz), but the same conclusion is valid for a combination of waves. In the far field the z directed magnetic potential From this point, the original cross section is relevant only for calculating the equivalent separation vector d in the twin lead vector Az is expressed as representation, and the remaining calculation bases solely on 2 z ′ ′ −jkz this twin lead representation. In appendix B we show examples Az = µ0 dz dc Kz(c)e G(R) (A.1) Zz1 I for the calculation of the twin lead equivalent for given cross where the dz′ integral goes on the whole length of the TL, sections. Using the twin lead representation, Eq. (A.7) may be e−jks G(s)= (A.2) rewritten 4πs −jkz + is the 3D Green’s function, is the surface current distri- Az =µ0e I jk[dx cos ϕ + dy sin ϕ] Kz ′′ ′′ bution as function of the contour parameter c (i.e. c and c , z2−z e−jk[z +r(z )] ρ 1 2 ′′ (A.10) dz ′′ ′′ , see Figure 1) which is known from electrostatic considerations, Zz1−z 4πr(z ) r(z ) and R is the distance from the integration point on the contour of the conductors to the observer: where dx and the dy are the x and y components of the vector d. This represents a twin lead, as shown in Figure 2, and is R = (x x′(c))2 + (y y′(c))2 + (z z′)2. (A.3) actually a 2D dipole approximation of the TL. Without loss p − − − 13 of generality, one redefines the x axis to be aligned with d, so that dx = d and dy =0, obtaining ′′ ′′ z2−z −jk[z +r(z )] −jkz + ′′ e ρ Az = µ0e I jkd cos ϕ dz ′′ , Zz1−z 4πr r(z ) (A.11) which is equivalent of having a current I+e−jkz confined on the conductor at x = d/2 and the same current confined on the conductor x = d/2 but defined in the opposite direction, representing a twin− lead (see Figure 2). We are interested in radiation, so we require the observer to be many wavelengths far from the TL: kρ 1 and kr 1, so that Eq. (A.11) may be further simplified≫ to ≫ ′′ ′′ + −jkz z2−z −jk[z +r(z )] µ0I e d cos ϕ ∂ ′′ e Az = dz ′′ . − 4π ∂ρ Zz1−z r(z ) (A.12) Fig. B.1. Panel (a) shows a cross section of circular shaped conductors and The dz′′ integral results in the exponential integral function panel (b) shows a cross section of rectangular shaped conductors. The sizes are in units of cm. The y axis is horizontal, and the x axis for each cross Ei as follows section is the symmetry axis. The red points show the current images which µ I+e−jkzd cos ϕ define the twin lead representation, and are referred further on. A = 0 z − 4π ∂ z2−z Ei jk z′′ + (z′′)2 + ρ2 , (A.13) ∂ρ − h p i z1−z where the Ei function satisfies d Ei(s)/ds = es/s. The twin lead geometry also allows us to use simple models for the termination currents in the x direction (see Figure 2), defining the x component of the magnetic vector potential, calculated as: d/2 + ′ −jkz1,2 Ax 1,2 = µ0I dx e G(R1,2) (A.14) ± Z−d/2 Fig. B.2. The magnitude of the electric field, the hottest color representing high field and coldest color low (close to 0) field, for the circular shaped where the indices 1,2 denote the contributions from the termi- conductors cross section in the left panel and for the rectangular shaped conductors in the right panel. The red scars in the middle of the plots show nation currents at z1,2, respectively, see (see Figure 2) and the the coordinate’s origin. distances R1,2 of the far observer from the terminations may be expressed in spherical coordinates, as follows: ′ R r z cos θ x sin θ cos ϕ, (A.15) field on the conductors. Given the surface currents Kz(c1) > 0 1,2 ≃ − 1,2 − and Kz(c2) < 0 in Eq. (A.9), and using the field intensity The integral (A.14) is carried out for kd 1, resulting in ≪ which is positive, E(c1) is proportional to Kz(c1) and E(c2) + −jkz1,2 (1−cos θ) is proportional to K (c ), we may calculate the separation Ax 1,2 = µ0I dG(r)e (A.16) − z 2 ± vector d, using Eqs. (A.8) and (A.9), after replacing Kz(c1) by E(c ) and K (c ) by E(c ) as follows APPENDIX B 1 z 2 − 2 COMPUTATION OF RADIATION PARAMETERS d dc1E(c1)ρ(c1) dc2E(c2)ρ(c2) To calculate the power radiated from a TL, of any cross = − . (B.1) H dc1EH(c1) section, one needs the separation vector d, defined in Fig- H ure 1, calculated from Eq. (A.9). If one needs the normalized For the cross sections in Figure B.1, the “positive” and radiation due to a forward wave, one also needs to know the “negative” conductors are symmetric, so that a given location characteristic impedance Z0. Those are obtained with the aid vector ρ(c2) on the negative conductor, is the minus of the of the ANSYS 2D “Maxwell” simulation, from an electrostatic corresponding location vector ρ(c1) on the positive conduc- analysis. We ran the 2D “Maxwell” simulation on two cross tor, and by symmetry the magnitudes of the electric fields sections shown in Figure B.1. It is to be mentioned that we E(c1) = E(c2), so that we may drop the second integral know the analytic solution for the cross section in panel (a) in the numerator of Eq. (B.1), and multiply the result by 2. from image theory [10], so that it can be used as a test Also by symmetry the y component of d comes out 0, so that for the quality of the numerical simulation. The magnitude d = d = dx represents the distance between the “image” of the electric fields measured for those cross sections is currents| | in the twin lead model, and we obtained d =2.54 cm shown in Figure B.2. We remark that the surface currents, for the circular cross section (compares well with Eq. (25)), which are proportional to the (tangential) magnetic field on and d = 2.91 cm for the rectangular cross section, as shown the conductors are also proportional to the (normal) electric in Figure B.1. 14

We obtained from the 2D “Maxwell” simulation also the well with Eq. (26)) and 99.51 Ω for the circular and rectangular per unit length capacitances, which came out 31.5 pF/m and cross sections, respectively. 33.51 pF/m for the circular and rectangular cross sections, The procedure described in this section can be done for any respectively. The characteristic impedance Z0 is calculated by cross section, and it supplies all the values needed to calculate 1/(Cc), where c is the velocity of light in vacuum and C is the the normalized radiation in Eq. (15). Its accuracy can be found per unit length capacitance, and come out 105.8 Ω (compares by comparing the values obtained for d and Z0 for the circular [2] R. Ianconescu and V. Vulfin, “Simulation and theory of TEM transmission cross section, with the theoretical values obtained from image lines radiation losses”, ICSEE International Conference on the Science of Electrical Engineering, Eilat, Israel, November 16-18, 2016 theory and they fit with an inaccuracy of less than 0.5%. [3] C. Manneback, “Radiation from Transmission Lines”, Transactions of the American Institute of Electrical Engineers, Vol. XLII, pp. 289-301, 1923 EFERENCES ; C. Manneback, “Radiation from Transmission Lines”, Journal of the R American Institute of Electrical Engineers, 42(2), pp. 95-105, 1923 [1] R. Ianconescu and V. Vulfin, “TEM Transmission line radiation losses [4] J. E. Storer and R. King, Radiation Resistance of a Two-Wire Line, in analysis”, Proceedings of the 46th European Microwave Conference, Proceedings of the IRE, vol. 39, no. 11, pp. 1408-1412, doi: 10.1109/JR- EuMW 2016, London, October 3-7 PROC.1951.273603, (1951) [21] R. Ianconescu and V. Vulfin, Analysis of lossy multiconductor trans- [5] G. Bingeman, “Transmission lines as antennas”, RF Design 2(1), pp. 74- mission lines and application of a crosstalk canceling algorithm, IET 82, 2001 MICROW ANTENNA P, 11(3), pp. 394-401 (2016) [6] T. Nakamura, N. Takase and R, Sato, “Radiation Characteristics of a [22] R. Ianconescu and V. Vulfin, “Simulation for a Crosstalk Avoiding Algo- Transmission Line with a Side Plate”, Electronics and Communications rithm in Multi-Conductor Communication”, 2014 IEEE 28-th Convention in Japan, Part 1, Vol. 89, No. 6, 2006 of Electrical and Electronics Engineers in Israel, Eilat, December 3-5 [7] JR. Carson, Radiation from transmission lines, Journal of the American [23] Brouwer L.E. “On continuous vector distributions on surfaces.(2nd com- Institute of Electrical Engineers 40(10), pp. 789-90 (1921) munication.)”, Koninklijke Nederlandse Akademie van Wetenschappen [8] Midlands State University, Zimbabwe, educational document: Proceedings Series B Physical Sciences. 12, pp 716-34 (1909) http://www.msu.ac.zw/elearning/material/1343053427HTEL104 [9] Collin, Robert E. Antennas and radiowave propagation, McGraw-Hill, 1985. [10] Orfanidis S.J., Electromagnetic Waves and Antennas, ISBN: 0130938556, (Rutgers University, 2002) [11] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Reuven Ianconescu graduated B.Sc (Summa cum Communication Electronics, 3rd edition, Wiley 1994 laude) in 1987 and PhD in 1994, both in Electrical [12] E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating engineering at Tel Aviv university. He worked many Systems, 2nd edition, Prentice Hall 1968 years in R&D in the hi-tech industry, did a post-doc [13] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd edition, John in the Weizmann Institute of Science, and is cur- Wiley & Sons, 2005 rently an academic staff member in Shenkar college [14] D. M. Pozar, Microwave Engineering, Wiley India Pvt., 2009 of engineering and design. His research interests are: [15] H. Matzner and K. T. McDonald, “Isotropic Radiators”, guided EM propagation, radiation and dynamics of arXiv:physics/0312023 charges and electrohydrodynamics. [16] R. Guertler, ”Isotropic transmission-line antenna and its toroid-pattern modification.” IEEE Transactions on Antennas and Propagation 25(3), pp 386-392 (1977) [17] A. Gover, R. Ianconescu, C. Emma, P. Musumeci and A. Friedman, “Conceptual Theory of Spontaneous and Taper-Enhanced Superradiance and Stimulated Superradiance”, FEL 2015 Conference, August 23-28, Daejeon, Korea Vladimir Vulfin has over 13 years of research expe- [18] R. Ianconescu, E. Hemsing, A. Marinelli, A. Nause and A. Gover, “Sub- rience in the areas of electromagnetic engineering. Radiance and Enhanced-Radiance of undulator radiation from a correlated Since 2004 he worked in different companies as electron beam”, FEL 2015 Conference, August 23-28, Daejeon, Korea Electromagnetics Engineer. His job duties included [19] A. Gover, R. Ianconescu, A. Friedman, C. Emma and P. Musumeci, design, simulations, implementations and measure- “Coherent emission from a bunched electron beam: superradiance and ments of antennas and microwave devices. Parallel stimulated-superradiance in a uniform and tapered wiggler FEL”, Nucl. to this, Vladimir was active as an external lecturer Instr. Meth. Phys. Res. A, http://dx.doi.org/10.1016/j.nima.2016.12.038 , and instructor in different universities and colleges 2016 in Israel. He lead more than 100 degree final projects [20] V. Vulfin and R. Ianconescu, Transmission of the maximum number of for students in Electrical Engineering Departments. signals through a Multi-Conductor transmission line without crosstalk or Vladimir received the B.Sc and M.Sc. degrees in return loss: theory and simulation, IET MICROW ANTENNA P, 9(13), Electrical Engineering. Currently he is working towards his Ph.D degree at pp. 1444-1452 (2015) Ben Gurion University in the area of antennas and metamaterials.